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To make the rarebit, place the cheese and milk in a heavy-based pan and season with salt and white pepper, to taste. Stir continuously over a medium heat to melt the cheese. Once melted, add the flour, breadcrumbs, and mustard powder, and continuously stir until it reaches a thick consistency. Pour onto a large flat tray and allow to cool to room temperature.
Once the mixture has cooled, tip into a food processor and blend. Add the whole eggs and egg yolks, one-by-one, to the food processor whilst blending. Add the Worcestershire sauce and pulse to just combine. Transfer into a piping bag and leave at room temperature until needed.
To make the yeast extract butter, mix the yeast extract into the soft butter in a bowl until fully incorporated, then add good squeeze of lemon juice. Set aside until needed.
To make the croque madame, preheat the oven to 200C/180C/Gas 5.
Brush both sides of each slice of sourdough with the flavoured butter. In a hot large frying pan, heat 50g/1¾oz od the unflavoured butter until foaming. Add 2 slices of the sourdough and fry until slightly light brown in colour. Turnover and repeat on each side, then set aside. Repeat with the remaining sourdough.
Pipe the rarebit onto one slice of bread, place 2 ham slices on topand a sundried tomato, followed by a slice of fried sourdough. Repeat to make three more sandwiches. Place the sandwiches on a baking try and bake until the cheese has melted and the sourdough is nicely toasted.
Meanwhile, heat a dash of vegetable oil in a frying pan and fry the eggs to your liking over a medium heat, ideally they should retain a runny yolk.
Remove each croque madame carefully out of the oven and place between 4 plates. Top with a fried egg, garnish with the gherkin halves, held in place with cocktail sticks.